The strongly regular (45,12,3,3) graphs

Using two backtrack algorithms based on dierent techniques, designed and implemented independently, we were able to determine up to isomorphism all strongly regular graphs with parameters v = 45, k = 12, λ = μ = 3. It turns out that there are 78 such graphs, having automorphism groups with sizes ranging from 1 to 51840

Equiangular lines in Euclidean spaces

We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d=14, and d=16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table

Simple inexpensive vertex and edge invariants distinguishing dataset strongly regular graphs

While standard Weisfeiler-Leman vertex labels are not able to distinguish even vertices of regular graphs, there is proposed and tested family of inexpensive polynomial time vertex and edge invariants, distinguishing much more difficult SRGs (strongly regular graphs), also often their vertices. Among 43717 SRGs from dataset by Edward Spence, proposed vertex invariants alone were able to distinguish all but 4 pairs of graphs, which were easily distinguished by further application of proposed edge invariants. Specifically, proposed vertex invariants are traces or sorted diagonals of $(A|_{N_a})^p$ adjacency matrix $A$ restricted to $N_a$ neighborhood of vertex $a$, already for $p=3$ distinguishing all SRGs from 6 out of 13 sets in this dataset, 8 if adding $p=4$. Proposed edge invariants are analogously traces or diagonals of powers of $\bar{A}_{ab,cd}=A_{ab} A_{ac} A_{bd}$, nonzero for $(a,b)$ being edges. As SRGs are considered the most difficult cases for graph isomorphism problem, such algebraic-combinatorial invariants bring hope that this problem is polynomial time.Comment: 6 pages, 4 figure

The strongly regular (45,12,3,3) graphs

Using two backtrack algorithms based on dierent techniques, designed and implemented independently, we were able to determine up to isomorphism all strongly regular graphs with parameters v = 45, k = 12, λ = μ = 3. It turns out that there are 78 such graphs, having automorphism groups with sizes ranging from 1 to 51840

The Pseudo-Geometric Graphs for Generalised Quadrangles of Order (3,t)

AMS classifications: 05E30; 51E12

Nonregular Graphs with Three Eigenvalues

We study nonregular graphs with three eigenvalues.We determine all the ones with least eigenvalue -2, and give new infinite families of examples.

5-Chromatic Strongly Regular Graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5.Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph.For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic.In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.

Non-geometric distance-regular graphs of diameter at least 33 with smallest eigenvalue at least −3-3

In this paper, we classify non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs with smallest eigenvalue at least $-3$, analogous to existing classification results available in the case that the smallest eigenvalue is at least $-2$